Regular coverings in complete modular lattices (Q1272086)

The Dedekind-Birkhoff Theorem states that in a finite-height modular lattice, every maximal chain has the same length and coverings in any two chains are in one-to-one correspondence. A covering \(\langle x,y\rangle \) in a modular lattice is upper regular if the joins satisfy: \(\bigvee y_i\) covers \(\bigvee x_i\) whenever they exist and \(i\) is a mapping from a chain \(I\) into \(\langle x,y\rangle \), written \(i\big (\langle x,y\rangle \big)=\langle x_i,y_i\rangle\), such that \(i<j\) implies \(\langle x_i,y_i\rangle \nearrow \langle x_j,y_j\rangle .\) The lower regular covering is defined dually. A covering \(\langle x,y\rangle\) is regular if it is both upper and lower regular. Main results: Theorem 1. Let \(L\) be a modular lattice. If \(L\) is of a finite height then all coverings are regular. If \(L\) is upper (lower) continuous then every covering in \(L\) is upper (lower) regular. Theorem 2. If \(L\) is a complete modular lattice then the regular coverings in any two maximal chains are in one-to-one correspondence.